Visual properties of generalized Kloosterman sums

TL;DR

This paper explores the complex visual patterns of generalized Kloosterman sums, revealing how specific number-theoretic conditions influence their geometric properties in the complex plane.

Contribution

It characterizes the number-theoretic conditions that produce distinct visual phenomena in generalized Kloosterman sums, extending understanding beyond classical sums.

Findings

Identification of conditions leading to specific visual patterns

Analysis of the complex plane distribution of sums

Insights into the differences from classical Kloosterman sums

Abstract

For a positive integer $m$ and a subgroup $\Lambda$ of the unit group $(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman sum is the function $K(a,b,m,\Lambda) = \sum_{u \in \Lambda}e(\frac{au + bu^{-1}}{m})$. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.